Systematic approach to equilibrium calculations

Using Excel Solver to solve the quadratic equation, p. 198 The systematic approach to equilibrium calculations mass balance and charge... 

Systematic approach to equilibrium calculations (mass and charge balance) Heterogeneous equilibria... 

Two principal features characterize the systematic approach to equilibrium calculations used in this book a) Expressing concentrations of every species by the product of a (a fraction of that species of all others in the same system. These fractions are a function of only the critical variable (e.g., pH), the relevant equilibrium constants, and C, the total concentration of the component, and b) Describing the equilibrium condition by a single balance equation, e.g., the proton balance equation (PBE), the ligand balance equation, etc. This results ultimately in a description of the equilibrium condition of the solution by one equation with a single concentration variable, i.e., in an implicit solution. 

The calculations involved in complex equilibria are the major subject of this chapter. The systematic approach to solving multiple-equilibrium problems is described. The calculation of solubility when the equilibrium is influenced by pH and the formation of complexes is also discussed. 

Chapter 6 The Systematic Approach to Equilibria Solving Many Equations Chapter 11 Complex Equilibrium Calculations Chapter 10 Complex Equilibrium Calculations... 

In summary, diagrammatic methods provide a systematic and powerful approach to the calculation of high-accuracy wavefunctions and energies, and offer considerable flexibility in the choice of the model. They are particularly valuable for small closed-shell molecules in near-equilibrium geometries but for larger systems, and for open-shell or quasidegenerate states, many problems are yet to be solved. 

This book systematically summarizes the researches on electrochemistry of sulphide flotation in our group. The various electrochemical measurements, especially electrochemical corrosive method, electrochemical equilibrium calculations, surface analysis and semiconductor energy band theory, practically, molecular orbital theory, have been used in our studies and introduced in this book. The collectorless and collector-induced flotation behavior of sulphide minerals and the mechanism in various flotation systems have been discussed. The electrochemical corrosive mechanism, mechano-electrochemical behavior and the molecular orbital approach of flotation of sulphide minerals will provide much new information to the researchers in this area. The example of electrochemical flotation separation of sulphide ores listed in this book will demonstrate the good future of flotation electrochemistry of sulphide minerals in industrial applications. 

Although free energies are related to equilibrium constants, it is concentrations that are either ultimately desired or experimentally determined. There are many ways to perform equilibrium calculations. We will approach such calculations through the extent of reaction, , defined in Eq. (5), because all concentrations can be expressed in terms of this single variable. In addition, use of the extent of the reaction allows us to perform equilibrium calculations in a very systematic manner. At equihbrium, the extent of reaction becomes the extent of reaction... 

If we wish to calculate the solubility of barium sulfate in a system containing hydronium and acetate ions, we must take into account not only the solubility equilibrium but also the other three equilibria. We find, however, that using four equilibrium-constant expressions to calculate solubility is much more difficult and complex than the simple procedure illustrated in Examples 9-4, 9-5, and 9-6. To solve this type of problem, the systematic approach described in Section llA is helpful. We then use this approach to illustrate the effect of pH and complex fonna-tion on the solubility of typical analytical precipitates. In later chapters, we use this same systematic method for solution of problems involving multiple equilibria of several types. 

You see that the same answer was obtained as when the problem was worked The systematic approach is applica-intuitively as in Example 6.4. You may think that the systematic approach is ex- ble to multiple equilibria, cessively complicated and formal. For this extremely simple problem that may be a justified opinion. However, you should realize that the systematic approach will be applicable to all equilibrium calculations, regardless of the difficulty of the... 

A different approach to a systematic study of the effects of experimental errors has also appeared. A computer program was developed which enabled ready calculation of the equilibrium constant from input experimental data. Beginning with synthetic data (no errors) small errors were deliberately introduced into the input data—for example, amounting to a weighing error of 0.3 mg in 20-500 mg, or an instrumental error of +0.003 absorbance units. Recalculation of the constant revealed that for certain concentration situations the determined equilibrium constant could be extremely sensitive to small experimental errors. The same conclusion was reached when K was determined by a graphical method with the same data. This again emphasizes the need for careful planning of experimental conditions. 

Calculations based on chemical equilibria tend to be somewhat complicated and often involve a large amount of data. So it is a good idea to approach these problems systematically. We ll illustrate how to handle some specific types of problems throughout this chapter. But first, it is worth noting three basic features of the strategy we ll employ in any equilibrium calculation. 

The equilibrium solubility of an Fe oxide can be approached from two directions -precipitation and dissolution. The first method involves precipitating the oxide from a supersaturated solution of ions with stepwise or continuous addition of base und using potentiometric measurements to monitor pH and calculate Fej- in equilibrium with the solid phase until no further systematic change is detected. Alternatively the oxide is allowed to dissolve in an undersaturated solution, with simultaneous measurement of pH and Fejuntil equilibrium is reached. It is essential that neither a phase transformation nor recrystallization (formation of larger crystals) occurs during the experiment this may happen with ferrihydrite which transforms (at room temperature) to a more condensed, less soluble phase. A discussion of the details of these methods is given by Feitknecht and Schindler (1963) and by Schindler (1963). 

Generally, we can calculate the hydrogen ion concentration or pH of an acid solution at equilibrium, given the initial concentration of the acid and its value. Alternatively, if we know the pH of a weak acid solution and its initial concentration, we can determine its K. The basic approach for solving these problems, which deal with equilibrium concentrations, is the same one outlined in Chapter 14. However, because acid ionization represents a major category of chemical equilibrium in aqueous solution, we will develop a systematic procedure for solving this type of problem that will also help us to understand the chemistry involved. 

The work of Reiss and co-workers puts the question of the equilibrium distribution of liquid embryos in dilute supercooled vapors on sound conceptual ground. However, having to calculate embryo free energies by simulation rules out the use of such an approach in practical applications. To overcome this limitation, Weakliem and Reiss [67] developed a modified liquid drop theory that combines elements of the physically consistent cluster with the conventional capillarity approximation. These same authors have also developed a rate theory which allows the calculation of nucleation rates in supercooled vapors [68]. The dependence of the predicted rates on supersaturation agree with classical nucleation theory, but the temperature dependence shows systematic deviations, in accordance with scaling arguments [54]. 

---